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The quadtree has recently become a major data structure in image processing. This correspondence investigates ways in which quadtrees may be efficiently stored as a forest of quadtrees and as a new structure we call a compact quadtree. These new structures are called virtual quadtrees because the basic operations we expect to perform in moving about within a quadtree can also be performed on the new representations. Space and time efficiency are investigated and it is shown these new structures often given an improvement in both.